Solving Exponential Equations: The Art of Simplifying Complex Expressions

Exponential equations can often seem daunting, but with the right techniques, they can be simplified and solved relatively easily. This article will guide you through the process of solving the equation 2x 21-x 3.

Understanding the Problem

The given equation is 2x 21-x 3. To solve this, we will use substitution to simplify the equation, making it easier to handle.

Substitution for Simplification

Let's start by rewriting the term 21-x in terms of 2x.

21-x 2 / 2x

Substituting this back into the original equation:

2x (2 / 2x) 3

Introducing the Variable 'y'

To simplify the equation further, let's introduce a variable y 2x. This substitution transforms the equation into:

y (2 / y) 3

Simplifying with Algebraic Manipulation

Multiplying both sides by y to eliminate the fraction:

y^2 - 2 3y

Reorganizing this equation to form a standard quadratic equation:

y^2 - 3y - 2 0

Factoring the Quadratic Equation

Factoring the quadratic equation:

(y - 1)(y - 2) 0

Setting each factor to zero gives us:

y - 1 0 → y 1

y - 2 0 → y 2

Solving for 'x'

Recall that y 2x. Thus, we can solve for x by equating y to these values:

For y 1: 2x 1 → x 0

For y 2: 2x 2 → x 1

Therefore, the solutions to the equation 2x 21-x 3 are:

x 0 and x 1

Conclusion

The process of solving exponential equations often involves using substitution to simplify the terms and then solving the resulting equations using algebraic methods. In this article, we demonstrated this with the example 2x 21-x 3, showing that the solutions are x 0 and x 1.